A chord of a circle of radius `10 cm` subtends a right angle at the centre. The area of the minor segments (given `pi=3.14)` is

Options

A. `32.5cm^(2)`
B. `34.5cm^(2)`
C. `28.5cm^(2)`
D. `30.5cm^(2)`

Explanation

To find the area of the minor segment of a circle with a radius of 10 cm, where a chord subtends a right angle (90 degrees) at the center, we will follow these steps:

### Step 1: Identify the radius and angle
- Given radius \( r = 10 \) cm and angle \( \theta = 90^\circ \).

### Step 2: Calculate the area of the sector
- The formula for the area of a sector is:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
- Substituting the values:
\[ \text{Area of sector} = \frac{90}{360} \times 3.14 \times (10)^2 \]

### Step 3: Simplify the fraction
- Simplifying \( \frac{90}{360} = \frac{1}{4} \):
\[ \text{Area of sector} = \frac{1}{4} \times 3.14 \times 100 \]

### Step 4: Calculate the area of the sector
- Continuing with the calculation:
\[ \text{Area of sector} = \frac{1}{4} \times 314 = 78.5 \text{ cm}^2 \]

### Step 5: Calculate the area of the triangle
- The triangle formed by the radius and the chord is a right triangle. The area of a triangle is given by:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, both the base and height are equal to the radius (10 cm):
\[ \text{Area of triangle} = \frac{1}{2} \times 10 \times 10 = 50 \text{ cm}^2 \]

### Step 6: Calculate the area of the minor segment
- The area of the minor segment is found by subtracting the area of the triangle from the area of the sector:
\[ \text{Area of minor segment} = \text{Area of sector} - \text{Area of triangle} \]
- Substituting the values:
\[ \text{Area of minor segment} = 78.5 - 50 = 28.5 \text{ cm}^2 \]

### Final Answer
- The area of the minor segment is \( 28.5 \text{ cm}^2 \).

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